Problem: Let $a_n = \frac{10^n-1}{9}$. Define $d_n$ to be the greatest common divisor of $a_n$ and $a_{n+1}$. What is the maximum possible value that $d_n$ can take on?
Solution: From the Euclidean Algorithm,
\[
\gcd(a_{n}, a_{n+1}) = \gcd(a_n, a_{n+1} - 10a_n).
\]We compute $a_{n+1} - 10a_n = \frac{10^{n+1}-1}{9} - \frac{10^{n+1}-10}{9} = 1$. Therefore, $a_{n+1}$ and $a_n$ share no common factors and $d_n$ is always $\boxed{1}$.